The scientific study of finite distortion of cubic metal crystals was formally inaugurated by G. I. Taylor's 1923 Bakerian Lecture to the Royal Society (Taylor & Elam 1923), and the development of a rational mechanics of finite plastic deformation of crystalline solids may be said to have begun. The remarkable feature of this pioneering work (and of all Taylor's subsequent experimental investigations of f.c.c. and b.c.c. crystals) is that, in addition to the use of X-ray analysis to determine the changing orientation of the crystal atomic lattice, external measurements sufficient to completely define the uniform distortion of the crystal specimen were made at each stage of the test. All material directions that remained unchanged in length were then established by exact geometric analysis.

Taylor's approach is to be distinguished from that of Mark, Polanyi & Schmid (1923), whose experimental study of crystals of hexagonal structure is of comparable historical importance. They pulled single crystal wires of zinc and assumed that slip lines (or bands) on the specimen surface were traces of a family of planes of single slip, which were then shown to coincide with a crystal plane. Taylor (1926), discussing such methods in general, rather amusingly remarked: “They depend in fact on knowing the form which the answer will take before starting to solve the problem.”

Taylor & Elam (1923) firmly established two experimental laws of fundamental and lasting value for the foundations of crystal mechanics that seven decades of experimentation on f.c.c. crystals have only served to reinforce.

As remarked in Chapter 1, v. Goler & Sachs (1927) derived equations for equal double slip in f.c.c. crystals in tension with the loading axis on a symmetry line, and Taylor (1927a) derived comparable equations in compression (and also gave the equation of the unstretched cone for f.c.c. crystals). Equations applicable to any crystal class for these same symmetry conditions and for both tension and compression were developed by Bowen & Christian (1965), who presented formulas for various specific combinations of slip systems in f.c.c. and b.c.c. crystals. A general equation for the deformation gradient in (proportional) double slip of arbitrary relative amounts was first given in the work of Chin, Thurston, & Nesbitt (1966) mentioned previously. They carried the analysis no further, however, and applied the equation only to cases of equal, symmetric double slip in f.c.c. crystals.

Apparently the first explicit equations for rotation and stretch of a crystal material line in arbitrary (proportional) double slip in f.c.c. and b.c.c. crystals were developed by Shalaby & Havner (1978) (independently of the work of Chin, Thurston, & Nesbitt (1966)). The equations were illustrated for various nonsymmetric axis positions and relative amounts of slip. General equations for material line and areal vectors and both the finite deformation gradient and its inverse in arbitrary (proportional) double slip were derived in Havner (1979). Here we shall follow this last approach to the analysis of double slip in crystals.

Rodney Hill wrote his preface to The Mathematical Theory of Plasticity 41 years ago this month. As a reader of the present monograph likely knows, that classic work dealt with the macroscopic theory of metal plasticity and its applications as the subject stood at mid-century; and Hill only briefly (albeit superbly) discussed in his introductory chapter the physical background of the plastic properties of crystals and polycrystalline aggregates. The same year, however, saw publication of the English translation of an earlier (1935) classic specifically concerned with that background, Kristallplastizität by E. Schmid and W. Boas. Not entirely coincidentally, both Rodney Hill and this translation were associated with the Cavendish Laboratory, Cambridge, during the period immediately following World War II.

Each of these books when first published was in many respects a treatise on its respective subject, but there was no contemporary work which integrated these fields. Today, I doubt a comprehensive treatise could be written on all that has transpired both in the development of mathematical theory and in the experimental study of plastic behavior of crystalline materials during this century (or even since 1950). Accordingly, in planning and carrying out the writing of the present work, I decided to restrict its scope to those aspects of the broad subject of crystalline plasticity that have particularly interested me and that I have contributed to or at least seriously studied during the years since 1968.

Introduction

In this chapter, we will be putting the results we proved in Chapter 6 to work. We will develop algorithms to solve a variety of optimization problems, all important in their own right.

The first is to minimize the cost of a network joining together several nodes. This can always be achieved by using what is called a ‘greedy algorithm’.

Another problem is to find the distance between any two nodes along a given network, say a road network. Two simple iterative algorithms exist for this problem. A related problem is to find the path of longest length between two vertices of an acyclic directed graph. This arises in certain types of sequencing problems, where the edges represent elapsed times. We shall see that one of our algorithms for shortest paths can easily be adapted to solve this problem.

A different type of problem is exemplified by the construction of a timetable, given simple compatibility constraints. We can model this by colouring the vertices of a graph, but the best we can achieve is a heuristic algorithm, not optimal but just reasonably efficient.

We should note that Warshall's algorithm can also be interpreted as a graphical algorithm to calculate strong components, but that was treated when we studied transitive closures.

Implementations of these algorithms in Modula-2 can be found in the appendices.